Actually Welcome to the concept of calculations of bills.
The bill calculation goes as equation,
y = $250 + n*($10)
here, n = number of lines,
1.) For n = 2 , Y = cost = 250 + 20 = $270
2.) for n = 4 , Y = $290
3.) for n = 6 , Y = $310
4.) for n = x, Y = $250 + x*($10)
for lastly 12 members we get as,
Y = 250 + (12)*(10) = 250+120 = $370
Consider the lengths of consecutive 1-2 blocks.
block 1 - 1, 2 - length 2
block 2 - 1, 2, 2 - length 3
block 3 - 1, 2, 2, 2 - length 4
block 4 - 1, 2, 2, 2, 2 - length 5
and so on.
Recall the formula for the sum of consecutive positive integers,
Now,
which means that the 1234th term in the sequence occurs somewhere about 1/5 of the way through the 49th 1-2 block.
In the first 48 blocks, the sequence contains 48 copies of 1 and 1 + 2 + 3 + ... + 47 copies of 2, hence they make up a total of
numbers, and their sum is
This leaves us with the contribution of the first 10 terms in the 49th block, which consist of one 1 and nine 2s with a sum of .
So, the sum of the first 1234 terms in the sequence is 2419.